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Properties

Label 2-30e2-4.3-c2-0-47
Degree $2$
Conductor $900$
Sign $0.712 + 0.701i$
Analytic cond. $24.5232$
Root an. cond. $4.95209$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.85 − 0.758i)2-s + (2.85 + 2.80i)4-s − 4.09i·7-s + (−3.14 − 7.35i)8-s − 0.984i·11-s + 14.8·13-s + (−3.10 + 7.58i)14-s + (0.253 + 15.9i)16-s + 9.80·17-s + 27.3i·19-s + (−0.746 + 1.82i)22-s − 18.3i·23-s + (−27.4 − 11.2i)26-s + (11.4 − 11.6i)28-s + 2.80·29-s + ⋯
L(s)  = 1  + (−0.925 − 0.379i)2-s + (0.712 + 0.701i)4-s − 0.585i·7-s + (−0.393 − 0.919i)8-s − 0.0894i·11-s + 1.13·13-s + (−0.221 + 0.541i)14-s + (0.0158 + 0.999i)16-s + 0.576·17-s + 1.44i·19-s + (−0.0339 + 0.0828i)22-s − 0.797i·23-s + (−1.05 − 0.431i)26-s + (0.410 − 0.417i)28-s + 0.0967·29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.712 + 0.701i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.712 + 0.701i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(900\)    =    \(2^{2} \cdot 3^{2} \cdot 5^{2}\)
Sign: $0.712 + 0.701i$
Analytic conductor: \(24.5232\)
Root analytic conductor: \(4.95209\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{900} (451, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 900,\ (\ :1),\ 0.712 + 0.701i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.254969686\)
\(L(\frac12)\) \(\approx\) \(1.254969686\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.85 + 0.758i)T \)
3 \( 1 \)
5 \( 1 \)
good7 \( 1 + 4.09iT - 49T^{2} \)
11 \( 1 + 0.984iT - 121T^{2} \)
13 \( 1 - 14.8T + 169T^{2} \)
17 \( 1 - 9.80T + 289T^{2} \)
19 \( 1 - 27.3iT - 361T^{2} \)
23 \( 1 + 18.3iT - 529T^{2} \)
29 \( 1 - 2.80T + 841T^{2} \)
31 \( 1 + 1.96iT - 961T^{2} \)
37 \( 1 + 44.8T + 1.36e3T^{2} \)
41 \( 1 + 17T + 1.68e3T^{2} \)
43 \( 1 + 54.8iT - 1.84e3T^{2} \)
47 \( 1 - 58.8iT - 2.20e3T^{2} \)
53 \( 1 - 89.2T + 2.80e3T^{2} \)
59 \( 1 + 67.3iT - 3.48e3T^{2} \)
61 \( 1 - 36.0T + 3.72e3T^{2} \)
67 \( 1 + 25.5iT - 4.48e3T^{2} \)
71 \( 1 - 120. iT - 5.04e3T^{2} \)
73 \( 1 - 69.8T + 5.32e3T^{2} \)
79 \( 1 + 32.2iT - 6.24e3T^{2} \)
83 \( 1 + 68.1iT - 6.88e3T^{2} \)
89 \( 1 - 67.8T + 7.92e3T^{2} \)
97 \( 1 - 1.16T + 9.40e3T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−10.06290599105452211406300689337, −8.859210724604920200441580949351, −8.307841534127307325428463325767, −7.44851877083579457887830894847, −6.57222001969155567056543868574, −5.64674778169045299598891832457, −4.03334398233188672087184861230, −3.34612335876511087804729194961, −1.88087293848875740875915499103, −0.76075073680342387301760450793, 0.930714381744197909447017099008, 2.21886743543080113764295340225, 3.44533167618746882693618165378, 5.03822544844119913376366562036, 5.82234354526333740681217402485, 6.72657457717024503351936240996, 7.51936672514113589680713750287, 8.577773121423551331597019543209, 8.957042680063566788833140383298, 9.901037276866869185302003620719

Graph of the $Z$-function along the critical line